RESEARCH ARTICLE A robotic fish caudal fin: effects of stiffness and motor program on locomotor performance

Transcription

1 56 The Journal of Eperimental iolog 215, Published b The Compan of iologists Ltd doi: /jeb RESERCH RTICLE robotic fish caudal fin: effects of stiffness and motor program on locomotor performance Christopher J. Esposito 1, James L. Tangorra 2, *, rooke E. Flammang 3 and George V. Lauder 3 1 Electric oat, General Dnamics, Groton, CT 06430, US, 2 Department of Mechanical Engineering, Dreel Universit, Philadelphia, P 19104, US and 3 Museum of Comparative Zoolog, Harvard Universit, Cambridge, M 02138, US *uthor for correspondence (tangorra@coe.dreel.edu) ccepted 5 October 2011 SUMMRY We designed a robotic fish caudal fin with si individuall moveable fin ras based on the tail of the bluegill sunfish, Lepomis macrochirus. Previous fish robotic tail designs have loosel resembled the caudal fin of fishes, but have not incorporated ke biomechanical components such as fin ras that can be controlled to generate comple tail conformations and motion programs similar to those seen in the locomotor repertoire of live fishes. We used this robotic caudal fin to test for the effects of fin ra stiffness, frequenc and motion program on the generation of thrust and lift forces. Five different sets of fin ras were constructed to be from 150 to 2000 times the stiffness of biological fin ras, appropriatel scaled for the robotic caudal fin, which had linear dimensions approimatel four times larger than those of adult bluegill sunfish. Five caudal fin motion programs were identified as kinematic features of swimming behaviors in live bluegill sunfish, and were used to program the kinematic repertoire: flat movement of the entire fin, cupping of the fin, W-shaped fin motion, fin undulation and rolling movements. The robotic fin was flapped at frequencies ranging from 0.5 to 2.4 Hz. ll fin motions produced force in the thrust direction, and the cupping motion produced the most thrust in almost all cases. Onl the undulator motion produced lift force of similar magnitude to the thrust force. More compliant fin ras produced lower peak magnitude forces than the stiffer fin ras at the same frequenc. Thrust and lift forces increased with increasing flapping frequenc; thrust was maimized b the 500 stiffness fin ras and lift was maimized b the 1000 stiffness fin ras. Supplementar material available online at Ke words: fish, caudal fin, fin ra, robotics, biomimetics, fleibilit, locomotion. INTRODUCTION The caudal fin of fishes has often been viewed b engineers as a relativel simple propulsive surface that is flapped b fish in a twodimensional motion that is an etension of the undulator wave generated b motomal bod musculature (arrett et al., 1999; lvarado and Youcef-Toumi, 2006; nton et al., 2009). However, in man fishes the caudal fin is used not eclusivel for thrust production, but also to create forces and moments that control the orientation of the fish via comple conformational changes and motion programs (Lauder, 1982; Drucker and Lauder, 1999; Lauder, 2000; Lauder and Drucker, 2004; Flammang and Lauder, 2008; Ttell et al., 2008; Flammang and Lauder, 2009). These shape changes and kinematic patterns are produced b fin ras within the caudal fin that are moved b intrinsic caudal musculature that is distinct from the segmented bod muscles (Lauder, 2006; Flammang and Lauder, 2008; Flammang and Lauder, 2009). on fish such as bluegill sunfish (Lepomis macrochirus) are thus able to activel deform the surface of their fins through individual fin ra motions and/or b altering the stiffness of individual fin ras (lben et al., 2007; Flammang and Lauder, 2008; Lauder et al., 2011a; Lauder et al., 2011b). The caudal fin of bon fish such as the bluegill sunfish has been demonstrated to generate locomotor forces in the lateral, lift and thrust directions, indicating that the tail is being used for more than just propulsion (Lauder and Madden, 2006; Lauder and Madden, 2007; Flammang et al., 2011). Fish are capable of altering the relative magnitudes of these forces and to vector water momentum in appropriate directions to eecute maneuvers; this abilit is a function of the design of the caudal fin with its individuall controllable fin ras. Historicall, fish-inspired robots that have modeled the caudal fin have used a rigid or moderatel fleible flat surface that is actuated about a single ais in an undulator or flapping motion from the caudal peduncle area (where the bod narrows and transitions to the tail). Studies like these have used the caudal fin to create forward propulsion and to control simple turning maneuvers b biasing the flapping of the fin to one side of the robot. This approach has proven useful for a number of fish-inspired robotic designs (arrett et al., 1999; lvarado and Youcef-Toumi, 2006; Long et al., 2006; Zhang et al., 2008; nton et al., 2009; Long et al., 2010; Low and Chong, 2010). However, a two-dimensional flatplate-like representation of the three-dimensional (3-D) fin motion without the abilit to activel control tail conformation means that it is difficult for current robotic designs to control the direction and magnitude of the force vector. Recent computational studies, such as those b Zhu and Shoele (Zhu and Shoele, 2008; Shoele and Zhu, 2009), have begun to address the role of fin ras, ra stiffness and fin conformational changes on force production, but there are no eperimental data from caudal fin robotic devices that allow the evaluation of ke parameters such as stiffness, motion program and frequenc on locomotor force production.

2 Caudal fin biorobotics 57 D C E F FR1 FR2 FR3 FR4 FR5 FR6 Fig. 1. Overview of the biomimetic design of the robotic fish caudal fin. () luegill sunfish, Lepomis macrochirus, the biological model for this robotic device. () Cleared-and-stained and (C) computed microtomograph scan of the sunfish tail region. ones are stained red in. (D) Overview of the caudal fin robot held on a stand in the laborator. The frame houses the rotational servomotors, controller board and attachment mechanisms for the carriage sstem. The low-stretch string tendons, attached to the servomotor horns, are guided through the bod to the fin ras. (E) View of the caudal fin robot shown in the flow tank where eperiments were conducted. The low-friction air carriage on which the robotic device is mounted is out of view at the top. (F) View of the caudal fin robot with si fin ras protruding from the peduncle of the robot. The fin ras are labeled 1 through 6, with 1 considered b our convention the dorsal-most fin ra, and 6 the ventral-most fin ra. The webbing and tendons are removed to show the peduncle area and fin ra attachment mechanism. The biorobotic fin was oriented in the horizontal plane to facilitate eperimentation. Lift forces were therefore measured in the horizontal plane (corresponding to the dorsoventral ais of a fish, in the -ais orientation) and thrust forces were measured along the rostrocaudal ais of the fish (along the -ais). The goal of this paper is to contribute to our understanding of fish caudal fin function b constructing a robotic model with si individuall controllable fin ras that allows detailed control over the movement pattern of the fin. The robotic caudal fin was programmed to move in five distinct patterns based on caudal fin kinematics measured in our previous work on freel swimming bluegill sunfish (Flammang and Lauder, 2008; Flammang and z Lauder, 2009). The caudal fin robot was then used to investigate the effect of changing fin ra stiffness, flapping frequenc and kinematic pattern on locomotor performance as measured b patterns of thrust and lift produced during the tail flapping ccle. MTERILS ND METHODS Design of the caudal fin robot si fin-raed robotic caudal fin was developed using information obtained from biological studies of the bluegill sunfish caudal fin (Fig. 1) (Lauder, 1982; Lauder, 2000; Lauder and Drucker, 2004; Flammang and Lauder, 2008). This design process was similar to that used for the robotic pectoral fin b developed in Tangorra et al. (Tangorra et al., 2007; Tangorra et al., 2010). The robotic caudal fin model was scaled to have linear dimensions approimatel four times larger than an adult bluegill sunfish of 20 cm total length. The 19 fin ras of the biological caudal fin were modeled using si engineered fin ras. This was the minimum number of ras required to satisfactoril eecute the desired kinematic patterns, and allow independent control of the dorsal and ventral tail lobes. This fin ra number was also sought to limit the compleit of the design and to help reduce the potential for failure. Each of the robotic fin ras were individuall actuated b a rotational servomotor (HS-645MG, Hitec RCD, Powa, C, US) via low-stretch polethlene tendons attached to the base of the fin ra. The compliant webbing material covering the fin ras was a blend of polester (82%) and elastane (18%) (Under rmour, Inc., altimore, MD, US), which was coated with a thin laer of late to decrease fluid diffusion and to stiffen the material slightl (Fig. 2). The bod supporting the robotic caudal fin tapers smoothl toward the caudal peduncle (Fig. 1D,E) to reduce total drag and minimize the turbulent effects of an protuberances that could alter water flow over the robotic tail. The region of the caudal peduncle was covered with a cloth sleeve (Fig. 2) that served to smooth this transition area and cover the fin ra bases. Tendons to control the fin ras emerged from slots at the posterior end of the bod (Fig. 2) and were attached to each side of the fin ra bases (Fig. 2C). The fin ra bases allowed fin ras of different stiffness to be attached without altering the tendon tension. Tensioning of the individual fin ra tendons was an important step in setting up the robotic caudal fin, and setting the correct tendon tension proved important in being able to produce the desired motion. To aid in this process, small tensioners were custom designed (Fig. 2D) and were located above the water level to allow individual tendons to be tightened or loosened as needed. Tendons from the servomotors located above the water entered the bod through slots, and made a 90 deg turn inside the bod around low-friction blocks (Fig. 2E) to emerge at the base of each fin ra. Kinematics Five movement patterns derived from three-dimensional kinematic data and high-speed video of the caudal fin of bluegill sunfish during stead swimming and maneuvering, from previous studies b Flammang and Lauder (Flammang and Lauder, 2008; Flammang and Lauder, 2009), were selected to be programmed into the robotic model. For ease of communication in this paper, these five motion programs are labeled as flat (supplementar material Movie 2), cupping (supplementar material Movie 3), W, undulation (supplementar material Movie 4) and rolling (supplementar material Movie 5) (Fig. 3). Our baseline motion program is designated as the flat motion program in which all fin ra bases were moved snchronousl through a tail beat with angular ecursions (Fig. 3, flat motion) so that all caudal fin ras moved together. Even though fluid loading

3 58 C. J. Esposito and others E D C Fig. 2. Specific features of the design of the robotic fish caudal fin. () View from above to show the low-stretch polethlene fin ra tendons entering through slots in the bod, the cloth boot that streamlines the caudal peduncle area with the transition to the fin ra bases, and the caudal fin (with black dots added for ease of 3-D kinematic tracking). () Fin ra bases (see Fig. 4) allow fin ras of different stiffness to be attached. Grooved slots allow the fin ra tendons to emerge from the bod interior. (C) Detail of fin ra bases with fin ras attached. Tendons from each side emerge and are tied to the fin ra bases. (D) Tensioners located above the water level allow fine adjustment of the tension in each fin ra tendon. (E) Interior view of the bod to show the Teflon rods that serve as bearings to alter the tendon direction b 90 deg (tendons are visible as the thin white polethlene threads) from the entr angle into the bod to the direction needed for fin ra attachment at the caudal peduncle. resulted in some bending of ras so that the tail was not trul flat and plate-like, all ras were driven b the same motion program and the base of tail was a uniform (flat) shape from dorsal to ventral. Three of the other four motion programs used the same ecursion amplitudes as the flat motion, whereas the rolling motion had differential fin ra amplitudes. The cupping, undulation and W motions were created b changing the phase relationships between the fin ras as described below. For these motions, maimum transformation in fin shape developed as the fin passed through the mid-sagittal plane. The motor trajectories were chosen such that the bases of the fin ras moved through a ±27 deg sweep. This sweep angle was an average determined from a 3-D analsis of stead swimming caudal fin kinematics of bluegill sunfish. The cupping motion of the tail is frequentl observed in live bluegill (Flammang and Lauder, 2008; Lauder and Drucker, 2002; Ttell, 2006), and is generated b having the dorsal- and ventralmost fin ras lead the tail beat (Fig. 3, cupping motion). This results in the center of the tail lagging behind during lateral motion and producing a cup-like shape of the tail surface. The cupping motion was smmetrical about the midline. The middle fin ras lagged the dorsal and ventral fin ras b 25 and 50 deg (out of a 360 deg ccle). The W-shape motion was produced when the two middle caudal fin ras, in addition to the dorsal- and ventral-most ras, led the tail beat. This generated a W-shape of the tail trailing edge (best seen in posterior view) with separate cupping areas in both the dorsal and ventral lobes of the fin (Fig. 3, W motion). This motion, which was smmetrical about the midline, applied phases of 0, 33 and 5 deg, to fin ras FR1, FR2 and FR3, respectivel, in the dorsal tail lobe, and corresponding phases in the ventral lobe. The undulation motion was generated b a dorsoventral phasedisplaced wave of activation of the fin ras through the tail beat, producing a wave of bending that travels down the fin from dorsal to ventral, a direction orthogonal to the flapping motion (Fig. 3, undulation motion). For the undulation motion, phases of 0, 15, 30, 50, 65 and 75 deg, from the dorsal-most fin ra to the ventral-most fin ra, were applied. Finall, the rolling motion was produced b having the fin ras undergo differential ecursions so that there was a smooth reduction in ecursion amplitude from dorsal to ventral among the si fin ras (Fig. 3, rolling motion). Ventral fin ras thus underwent much smaller side-to-side ecursions than the upper fin ras, in contrast to the other four motion programs where all si fin ras moved through the same ecursion, while preserving a zero phase lag. During this motion, the sweep of the fin ras decreased linearl from the dorsal-most fin ra to the ventral-most fin ra, with a maimum sweep of ±27 deg and a minimum of ±5 deg. Mechanical properties of fin ras The fleural rigidities the modulus of elasticit (E) times the area moment of inertia (I) (Gere, 2004) of the bluegill sunfish fin ras were determined b conducting three point bending test on freshl dissected fin ras. Methods for these tests are provided in lben et al. (lben et al., 2007) and Lauder et al. (Lauder et al., 2011a; Lauder et al., 2011b). Several points along the length of each caudal fin ra were tested b measuring the applied force and the resulting displacements. First-order Euler ernoulli equations were applied to data acquired from the three point bending tests to determine the passive fleural rigidities along fin ra lengths. This calculation ignored small changes in the width of the fin-ra over the approimatel 2 mm test area. Computed microtomograph (voel dimensions: mm, mm, z 0.12 mm) scans of a bluegill sunfish caudal fin (Fig. 1C) were used to estimate I along the lengths of different caudal fin ras. pproimatel ever 1.5 mm along the length of the fin ra, the microtomograph slices were eamined and the bone structure was outlined. To distinguish the edge of the bone structure from collagenous material, a 5 5 convolute filter was applied to each slice of the scan. To calculate the I of the bones, the outlines were converted to binar and imported to a MTL program (v R2008a; The MathWorks, Inc., Natick, M, US). custom MTL program was developed to determine the bending ais and calculate the second area moment of inertia. The fleural rigidities for the robotic fin ras were set so that one scaling of fleural rigidit values would be epected to behave and bend in a fashion similar to the biological fin ras, and other values were set to be more and less stiff to allow for an eploration of the stiffness parameter space and to determine the effect of fin ra stiffness on locomotor forces. To estimate the appropriate scaling for the robotic fin ras, fin ra bending was modeled using firstorder cantilever ernoulli Euler equations. This ignored the inertial response of the fin ras as the fin was accelerated. The hdrodnamic loading of the fin was determined from drag and added mass forces

4 Caudal fin biorobotics 59 Flat motion Cupping motion C Fig. 3. Motion programs for the robotic caudal fin were based on observed sunfish caudal fin kinematics (see Lauder, 2000; Flammang and Lauder, 2008; Flammang and Lauder, 2009). () Posterior views of a bluegill sunfish performing stead swimming and maneuvering motions. () Posterior views of the robotic caudal fin performing similar caudal fin movements. Images of the robotic caudal fin were rotated 90 deg for comparison. Yellow arrows in and indicate the direction of motion of the individual fin ras. (C) The driven motion of the si robotic fin ras during two complete ccles (each fin ra motion is plotted in a different color and dash pattern); From top to bottom, fin ra 1 (blue solid), fin ra 2 (green solid), fin ra 3 (red solid), fin ra 4 (blue dash), fin ra 5 (green dash) and fin ra 6 (red dash). W motion Undulation motion Rolling motion mplitude (deg) z Time (s) applied to the fin webbing. The width of the fin webbing was assumed to increase linearl along the fin s length. s a result, the load from drag increased along the length of the fin with the square of the flapping frequenc and with the cube of the distance from the peduncle. The load from the added mass increased along the length of the fin linearl with the fin s angular acceleration and with the fourth power of the fin s length. This assumed that each strip of the fin accelerated a volume of water that was equal to a section of a half clinder with a diameter equal to the fin s width. scaling factor of approimatel 500 was estimated to be appropriate for robotic fins that had linear dimensions four times those of a biological fin and that were to be flapped between 0.5 and 2.0 Hz.

5 60 C. J. Esposito and others C D E F 2.0 mm 500 fin ra G 2.5 mm 500 fin ra deforming cubic polnomial load Total length: 99.5 mm 0.75 mm 0.75 mm Fig. 4. Design of fin ras for the robotic caudal fin. ( E) The scaled robotic fin ras taper in height and width along their length. Each of the fin ras is labeled with the corresponding scaled stiffness value. The base crosssectional area and slope of the taper are changed to match the scaled fleural rigidit of the fish fin ras along their length. (F) Schematic of the 500 fin ra with no load applied to it. (G) cubic polnomial load is applied, showing the bending curvature of the 500 fin ra. The fleural rigidities of the biological fin ras were then multiplied b this scaling factor, and the cross-sectional geometries of the robotic fin ras were determined b fitting the fleural rigidities of the robotic fin ras to the scaled biological values. The robotic fin ras were tapered in both width and height along their lengths so that their bending characteristics approimated those of the biological fin ras as seen in vivo in previous 3-D kinematic studies of swimming bluegill sunfish (Figs 4, 5) (Flammang and Lauder, 2008; Flammang and Lauder, 2009). Sets of robotic fin ras were manufactured with fleural rigidities 150 to 2000 times those of the biological fin ras (Fig. 4 E). This range encompassed both ver stiff (2000 ) and ver compliant (150 ) fins. further discussion of fin ra scaling to match the material properties of manufactured fin ras to those of the bluegill sunfish is given in Tangorra et al. (Tangorra et al., 2010). The fin ras were manufactured using fused deposition modeling (Stratass Inc., Eden Prairie, MN, US; material: S plastic) and stereolithograph (3D Sstems, Rock Hill, NC, US; material: ccura 40 UV resin), and were approimatel 10 cm long, ecluding the attachment base (Fig. 4). Eperimentation Eperiments were conducted to determine how forces and flows were affected b alterations in fin ra stiffness, kinematic patterns and fin-beat frequenc. Five fins with fleural rigidities scaled 150, 250, 500, 1000 and 2000 times the fleural rigidities of the biological fin ras were tested. For each of the five stiffnesses, five motions were tested: flat, undulation, cupping, W and rolling. The fin was actuated at frequencies of 0.5, 1.0, 1.5, 1.8, 2.0, 2.2 and 2.4 Hz. The fin was mounted above a recirculating flow tank used for previous eperiments on both live bluegill and robotic pectoral fins (Lauder et al., 2007; Flammang and Lauder, 2008; Flammang and Lauder, 2009; Tangorra et al., 2009; Tangorra et al., 2010), and a flow speed of 100 mm s 1 was used for all tests. The robotic caudal fin was suspended in the flow tank (Fig. 1E) from a low-friction air-bearing sstem mounted above the tank. This allowed the assembl to be attached to two force transducers that measure forces in the horizontal plane (for details, see Lauder et al., 2007; Tangorra et al., 2010; Lauder et al., 2011a). The robotic fin was oriented horizontall (Fig. 1), as if a fish was ling on its side. The forces measured in the horizontal plane, therefore, equated to thrust (along the rostrocaudal -ais) and lift (along the dorsoventral ais, which we will refer to as the -ais, as in the same orientation for a live, swimming fish). The forces generated b the robotic caudal fin were measured using two Futek LS200 S-beam load cells (44.5 N, LS200, Futek Inc., Irvine, C, US) and sampled at 200 Hz (6036E data acquisition card, National Instruments, ustin, TX, US). Data were processed using a low-pass filter to reduce noise. The filter used a Kaiser window with a passband frequenc of 10 Hz, a stopband frequenc of 12 Hz and a peak error of 10 3 (Oppenheim et al., 1989). z z luegill Robot Fig. 5. Images from high-speed video to show the comparison between the motion of the bluegill sunfish caudal fin (upper) and that of the robotic caudal fin (lower) discussed in this paper. Two representative times are shown 180 deg out of phase to illustrate caudal fin deformation during locomotion. The robotic caudal fin is driven with a stiff bod, in contrast to the deforming bluegill bod, and robotic fin ras are actuated at joints with the caudal peduncle. Images have been scaled to the same approimate size; the actual length of the robotic tail is four times that of the bluegill tail. Similar curvatures between the robotic caudal fin and the bluegill tail are achieved with a scaled fin ra stiffness of 150 shown here. See supplementar material Movie 1.

6 Caudal fin biorobotics Force (N) Frequenc (Hz) Fig. 6. Mean () thrust and () lift produced b the 1000 fin as flapping frequenc was changed. Forces are shown for the flat (dark blue), cupping (green), W (purple), undulation (red) and rolling (light blue) motions. Means are based on five consecutive ccles and error bars represent ±1 s.d. Force ccles for analsis were calculated b overlaing and averaging five consecutive stroke ccles. To reduce transients that are associated with startup, the first five fin-beat ccles were discarded. The collection and analsis process of the force measurements is further discussed in Tangorra et al. (Tangorra et al., 2010), as the procedures in the present stud followed closel those used for our previous eperiments on robotic pectoral fins. Critical values for differences in thrust production were determined through binomial tests (Zar, 1999). Three high-speed (500 Hz), high-resolution (1 megapiel) cameras (Photron Fastcam PCI-1024 US, Inc., San Diego, C, US) were positioned to capture the dorsal, lateral and posterior views of the fish and robot in the flow tank simultaneousl with the force measurements, as in our previous research (Drucker and Lauder, 2005; Lauder et al., 2007; Lauder and Madden, 2007; Standen and Lauder, 2005; Standen and Lauder, 2007; Tangorra et al., 2010). These data provided 3-D kinematic views of the moving caudal fin, and calibration b direct linear transformation (Hedrick, 2008) allowed quantification of robotic caudal fin motion and in vivo fin ra bending. Hdrodnamic wakes of the robotic caudal fin were studied using digital particle image velocimetr (DPIV), as in previous research (Drucker and Lauder, 1999; Lauder, 2000; Nauen and Lauder, 2002; Tangorra et al., 2010). riefl, light from a continuous 8 W argonion laser was focused onto a light sheet that was projected (in separate eperiments) both parallel to the plane of the tail and orthogonal to the tail plane. This allowed us to capture images of wake flow patterns generated b the robotic fin for comparison with those produced from freel swimming bluegill sunfish in previous eperiments (Tongorra et al., 2010). DPIV data were collected simultaneousl with force measurements. RESULTS Effects of changing the fin motion pattern The force patterns generated b the robotic caudal fin are separated into two motion program groups smmetrical and asmmetrical to better eplain how force is associated with fin motions. Forces from the smmetrical motions (flat, cupping and W) were primaril in the thrust direction, whereas forces from the asmmetrical motions (undulation and rolling) had both significant lift and thrust components (Fig. 6). Values reported below are means ± 1 s.d. Forces from the smmetrical motions The forces from the three smmetrical motions flat, cupping and W were primaril in the thrust direction. lthough lift was produced, the magnitude of lift was ver small. In all smmetrical cases, the average magnitude of the lift force over a ccle was less than 4% of the average magnitude of the thrust. In general, as the fin accelerated, fluid was pushed along the rostrocaudal ais of the fin and thrust forces developed with the fin s angular velocit. The magnitude of the thrust increased as the fin accelerated from its etreme lateral position towards the midline, with a peak being reached approimatel when the fin reached mid-stroke (Fig. 7). The timing of the maimum peak was dependent on fin stiffness and the actuation mode, but in all trials, the peak thrust force occurred approimatel when the base of the fin passed through the midline. Often, small dips in the thrust force occurred before the maimum peak thrust force was reached (Fig. 7). Thrust began to graduall decrease after the base of the fin reached its maimum velocit, which was when it passed through the midline. s the fin reversed direction, the gradual decrease in thrust turned into a net drag force (seen where the red curve dips below the zero force line in Fig. 7). These results were consistent with previous results from studies that used computational fluid dnamics (CFD) to predict the forces generated b fleible, trapezoidal foils (khtar et al., 2007; Zhu and Shoele, 2008). lthough the force profiles created b the cupping motion were similar to those produced b the flat motion, the subtle changes that were made to the fin ra motions resulted in slight, but consistent, differences to the force profile. The differences between the magnitudes of the mean thrust forces were most pronounced when the fin was flapped at lower flapping frequencies and had higher fin stiffnesses; consequentl, when there was little bending of the fin ras the conformations of the cupping and flat fins were most different. For eample, when the fin was flapped at 1.0 Hz with a fin stiffness of 2000, the flat, cupping and W motions produced magnitudes of mean thrust of 0.190±0.001, 0.203±0.001 and 0.198±0.001 N, respectivel. This is a 7% difference between the flat and cupping, a 4% difference between the flat and W and a 3% difference between the cupping and W motion. When the fin used ver compliant fin ras (150 and 250 ) and/or the flapping frequenc was high (1.8 Hz and 2.0 Hz), the forces produced b the

7 62 C. J. Esposito and others Normalized time (s) Normalized force (N) Normalized force (N) C D Vorticit (s 1 ) Vorticit (s 1 ) Fig. 7. (,) Normalized force traces and (C,D) digital particle image velocimetr (DPIV) images of associated wake flows, during one ccle, for the flat (,C) and undulation (,D) motions of the robotic caudal fin. In the force traces, the red line indicates the thrust force and the blue line indicates the lift force. The graphs have been normalized to show the pattern produced b the motion. In the flow images, red represents a positive (counterclockwise) rotation of fluid and blue/purple represents a negative (clockwise) rotation of fluid. Yellow arrows indicate the velocit and direction of the fluid. The green region indicates an area of zero vorticit. The free-stream flow is moving at 100 mm s 1 in the positive -direction. robotic caudal fin varied much less. t a flapping frequenc of 1.8 Hz and a fin stiffness of 150, the W and flat motions produced mean thrust forces of nearl the same magnitude. The cupping motion produced a magnitude mean thrust force that was 5% greater than that of the flat and W motions. The mean forces created b the W motion were not significantl different than the forces created b the flat motion when the fin fleibilit was low (150 and 250 ; P<0.05) and/or when flapping frequenc was high (1.8 Hz and 2.0 Hz; P<0.05). The forces were consistentl greater when fin ra stiffness was increased and/or flapping frequenc was decreased. These results are a function of multiple driving inputs and will be discussed in detail in later sections. The mean thrust forces generated b the cupping motion were larger than those generated b both the smmetrical and asmmetrical motions during 84% (21 out of 25; P<0.001) of comparable trials. The cupping motion generated mean thrust forces that were 14±6% larger than those generated b the flat motion during comparable trials (Fig. 6). The cupping motion also produced mean thrust forces that were 8±3, 22±11 and 50±12% larger than those of the W, undulation and rolling motions. The three eceptions to this occurred during the W motion at a fin stiffness of 150 and a flapping frequenc of 2.0 Hz, and the W motion with a fin stiffness of 500 and a flapping frequenc of 1.8 and 2.0 Hz. In these three cases, the average magnitude of the thrust force produced b the W motion was less than 4% higher than that of the cupping motion. Forces from the asmmetrical motions In contrast to the small mean lift forces created b the smmetric motion programs, the asmmetric motions created lift and thrust forces where lift was of the same magnitude as thrust (Fig. 6). The thrust force profiles generated b the asmmetrical motions had a similar pattern to those of the flat motion (Fig. 7). These results are consistent with those found b Lauder (Lauder, 2000), who identified that the asmmetrical motions created b the bluegill sunfish during stead swimming show both a thrust and a lift force during stead swimming. The undulation motion generated thrust force as the dorsal-most fin ra was accelerated from the etreme lateral position towards the midline (Fig. 7). This thrust force graduall decreased and became a net drag force (Fig. 7, where the red line dips below the zero line) once the ventral-most fin ra passed the midline and the dorsal-most fin ra began its reversal. Lift forces were generated when all the fin ras moved in the same direction, from one etreme lateral position to the other. There was a gradual decrease in lift force when the dorsal-most fin ra reversed its direction. This gradual decrease led to small negative lift forces being generated during the reversal period (Fig. 7, where the blue line dips below the zero line). Lift forces were generated again when all the fin ras began moving in the same direction, after the reversal of the last (ventral-most) fin ra. The phase difference between the thrust and lift peak magnitudes was on average 38±4 deg. The variance is due to the bending of the fin ras. The undulation motion generated lift forces that were of the same magnitude as the thrust forces and had two-dimensional force vectors that were, on average, larger than those created b all other motions. t flapping frequencies above 0.5 Hz, lift forces generated b the undulation motion were on average 27±3% lower than the thrust forces. t a flapping frequenc of 0.5 Hz, the lift forces were significantl larger than the thrust forces. The magnitude of the mean thrust force generated b the undulation motion was on average 24±10% smaller than the magnitude of the mean thrust force generated b the flat motion. On average, the two-dimensional magnitudes of the undulation force were 48±6% higher than all other motions (48±1, 41±1 and 52±1% higher than the flat, cupping and W motions, respectivel). The rolling motion generated thrust force during the part of the sweep when the fin ras accelerated towards the midline from the etreme lateral position. fter the fin passed the midline, the fin ras began to decelerate and to produce less thrust force. During a portion of the stroke between the point when the fin passed the midline and when the fin reversed its stroke, the thrust turned into a net drag force. Lift was created during the rolling motions when

8 Caudal fin biorobotics 63 the fin ras accelerated from the midline to the etreme lateral position. Negative lift was produced during the part of the sweep when the fin accelerated from the lateral-most position to the midline. The peak lift force occurred after the thrust peaks with an average phase difference of 155±5 deg. The rolling motion generated thrust forces that were the smallest of all the smmetrical and asmmetrical motions, and mean lift forces that were onl slightl positive. Lift forces during the stroke were onl slightl larger than the negative lift forces generated, which caused the mean magnitude of lift to be onl slightl positive. Thrust forces generated were of the same magnitude as those of the flat motion. Lift forces produced during the rolling motion had both large negative and positive magnitudes ( 0.49 and 0.62 N, respectivel) This caused the mean lift force to never be higher than 0.054±0.002 N, which was onl N larger than the smallest mean lift force generated b the undulation motion. The mean thrust forces generated b the rolling motion were on average 28±12% smaller than those generated b the flat motion and 21±7% smaller than those produced b the undulation motion. Wake flows The five robotic motions each produced different wake flows. The subtle kinematic differences between the three midline smmetrical motions resulted in onl slightl different wake flows. The large differences in amplitude and phase between the asmmetrical motions resulted in significantl different wake flows. The flat motion shed two tip vortices from the leading edges of the dorsal- and ventral-most fin ras (Fig. 7C). These vortices were counter-rotating and had approimatel equal vorticit. The region between the vortices had an accelerated jet of fluid that was directed awa from the robot along the midline. The cupping motion produced two tip vortices that were similar to those generated b the flat motion. The vortices formed at the leading edges of the dorsal and ventral-most areas of the fin were also counter-rotating, and had approimatel equal vorticit. The paths that the vortices translated along, however, were not parallel to the midline like the vortices shed b the flat motion. The cupping motion vortices were funneled inward towards the midline. This funneling caused an increase in fluid velocit between the vortices. The W motion shed four vortices as the fin passed the midline and through the DPIV viewing area. Two of the vortices developed along the leading edges of the dorsal- and ventral-most fin ras. These vortices were similar in size and structure to those created b the other smmetrical motions. Similar to the wake vortices generated b the cupping motion, the vortices were funneled inward towards the midline. Two other vortices developed along the leading edge area around the notch of the fin. These vortices were equal to each other in size and structure, but were smaller and had lower vorticit than the vortices that developed along the dorsaland ventral-most fin ras. The vortices that developed at the notch area were also funneled inwards as the translated awa from the fin. The area between the four vortices had an accelerated jet of fluid directed awa from the robot along the midline. The undulation and rolling motions produced wake vortices that were asmmetrical about the midline and significantl more comple than the smmetrical motions. oth the asmmetrical motions shed multiple vortices throughout the stroke. The undulation motion produced wake flows that translated along the midline in the positive lift direction. total of four vortices were created: two developed from the leading edge at the dorsal and ventral-most fin ras and two were created from the notch area (Fig. 7D). The two notch vortices were considerabl smaller, had lower vorticit and translated at a quicker pace than the vortices created from the leading edge at the dorsal- and ventral-most fin ras. These notch vortices were consumed b the flow downstream because of their small size. The dorsal-most fin ra, FR 1, was the leading fin ra in the pattern. The first vorte was created when the leading edge of the dorsalmost fin ra passed through the viewing area. This vorte translated at an angle of approimatel 25 deg off the midline in the lift direction (ventrall). The second set of vortices was created when the ventral-most fin ra, FR 6, passed through the viewing area. These vortices funneled towards the midline at a smaller angle than the dorsal vorte. These caused the area of accelerated fluid between these vortices to be directed in the lift direction (ventrall). The rolling motion generated three noticeable vortices: one from the dorsal lobe, one from the ventral lobe and one from the notch area. Each of the three vortices was of different size and had different vorticit. The notch vorte was considerabl smaller, had a lower vorticit and was quickl consumed b the flow downstream. The leading edge of the dorsal lobe shed the first vorte. ecause the dorsal-most fin ra had the largest displacement, the dorsal lobe vorte was the largest. This vorte translated along the direction of the flow, funneling towards the midline. The vorte from the ventral lobe of the caudal fin was the second largest vorte and the third to be shed. The ventral lobe vorte translated at an angle towards the midline. Further downstream, the ventral vorte interacted with the notch and the dorsal vorte. The area of accelerated flow between the vortices was directed simultaneousl along two paths, approimatel ±45 deg off the midline. Effects of altering fin ra stiffness The forces generated b the robotic fin during the smmetrical and asmmetrical motions changed as the stiffness of the fin was altered (Fig. 8). The forces generated b each of the motions were affected differentl b the stiffness of the fin, but general trends were apparent. The more compliant fins (150 and 250 ) produced lower peak magnitude forces than the stiffer fins (1000 and 2000 ). For eample, during the flat motion at 1.5 Hz, the 150 fin produced a peak thrust force with a magnitude of 0.391±0.010 N. During the same conditions, the 2000 fin produced a peak thrust force with a magnitude of 0.594±0.013 N (52±3% larger). The magnitudes of the peak and mean forces generated b the W motion were significantl affected b the stiffness of the fin and the flapping frequenc. The magnitude of the peak and mean thrust forces generated b the W motion at low fin stiffness and/or high flapping frequenc were similar to the forces generated b the flat motion. s stiffness was increased and/or flapping frequenc was lowered, the difference between the forces was more pronounced. t a fin stiffness of 150 and a flapping frequenc of 1.5 Hz, the mean thrust produced from the W motion was, on average, the same as that produced from the flat motion. s the stiffness of the fin increased, the magnitude of the mean thrust force created b the W motion increased at a faster rate than the flat motion. t a fin stiffness of 2000 and a flapping frequenc of 1.5 Hz, the W motion produced, on average, 15±4% more mean thrust than the flat motion. The mean thrust forces generated b the asmmetric and smmetric motion programs peaked at different fin stiffnesses, but on average, the largest mean thrust forces were produced b the 500 fin. During 80% (20 out of 25; P<0.001) of comparable eperimental trials, the 500 fin produced the largest mean thrust forces. The 500 fin, during the flat motion, produced on average 64±9% more mean thrust than the 150 fin, 26±7% more mean thrust than the 250 fin, 15±7% more mean thrust than the 1000 fin and 18±8% more mean thrust than the 2000 fin. These results

9 64 C. J. Esposito and others Force (N) Frequenc (Hz) Fig. 8. Mean () thrust () and lift forces generated b the undulation motion b 150 (blue), 250 (red), 500 (green), 1000 (purple) and 2000 (brown) stiffness fin ras. ars are grouped b frequenc. Means are based on five consecutive ccles and error bars at the peaks represent ±1 s.d. associated with each stiffness. are representative of the other motions; the eact percent differences in mean thrust force depended on multiple factors, but the trend was common for all the robotic caudal fin motions. Out of the five eperimental trials where the 500 fin did not produce the largest magnitude mean thrust, four were at a flapping frequenc of 0.5 Hz. The last eception was the W motion at a flapping frequenc of 1.8 Hz. The 1000 fin produced, on average, 1±4% more force than the 500 fin. The mean lift forces produced b the robotic caudal fin during the asmmetrical motions undulation and rolling peaked at a particular stiffness (Fig. 8). The 1000 fin, when actuated through the asmmetrical motions, produced the largest magnitude mean lift forces during 75% of comparable trials (si out of eight; P>0.05). On average, the 1000 fin produced magnitude mean lift forces that were 123±54% larger than the ver compliant 150 fin, 16±4% larger than the 500 fin and 37±21% larger than the ver stiff 2000 fin. The two eceptions happened at the 0.5 Hz flapping frequenc during both the undulation and rolling motions. t this low flapping frequenc, the 150 fin produced the largest mean lift force, which was onl 7% (0.003 N) higher than the 1000 fin. Effects of changing flapping frequenc s epected, when the flapping frequenc of the robotic fin was increased, the magnitudes of the forces generated during the five motions also increased. This trend held for all comparable eperiment trails. The robotic caudal fin performing the flat motion with a fin stiffness of 1000 generated mean thrust forces that are representative of the overall trend. With an increase in flapping frequenc from 1.0 to 1.5 Hz, the mean thrust increased b 227%. n increase in the flapping frequenc b an equal amount, from 1.5 to 2.0 Hz, increased the mean thrust b onl 128%. t higher flapping frequencies, the mean thrust force increased at a slower rate than at lower flapping frequencies. The eact increases in mean thrust force depended on the stiffness of the fin, the driven motion of the fin ras and the increase in flapping frequenc, but the trend was common for all the robot caudal fin motions and stiffnesses. The peak magnitude and mean forces generated b the W motion were significantl affected b alterations in the stiffness and the flapping frequenc of the fin. s previousl shown, the W motion produced on average the same force as the flat motion at low fin ra stiffnesses. s the stiffness of the fin ras increased, the mean thrust force created b the W motion increased at a faster rate than that created b the flat motion. similar result was also found if the fin stiffness was held constant and the flapping frequenc was increased. t a fin stiffness of 250 and a flapping frequenc of 1.0 Hz, the W motion outperformed the flat motion b 5±1%. When the flapping frequenc was increased from 1.0 to 2.0 Hz and the fin stiffness was held constant at 250, the W and flat motions produced on average the same mean thrust force. During both the rolling and undulation motions, the magnitude of the mean lift forces peaked depending on the stiffness and flapping frequenc. With a fin stiffness of 1000, the mean lift forces created b the undulation and rolling motions peaked at 1.8 Hz. t flapping frequencies higher than 1.8 Hz, the mean lift forces produced b the fin decreased. The more compliant fins (150, 250 and 500 ) peaked at slightl lower flapping frequencies ( Hz). The stiffer fin (2000 ) peaked at slightl higher flapping frequenc (2.0 Hz). DISCUSSION Robotic and biological caudal fins The robotic caudal fin was designed to allow for the production of a variet of motions that had been observed in previous studies of bluegill sunfish caudal fin during locomotion and maneuvering (Lauder, 2000; Flammang and Lauder, 2008; Flammang and Lauder, 2009). designing the fin with individuall controllable fin ras that could be varied in stiffness among eperiments, we were able to assess the effect of tail fin ra stiffness on locomotor performance, and to manipulate the motion program and to quantif patterns of thrust and lift force generation with changes in tail-beat frequenc. lthough previous functional studies of caudal fin locomotion in freel swimming fishes have been instrumental in contributing to our understanding of aquatic propulsion, the inabilit to manipulate and isolate ke biomechanical parameters has hindered our understanding of fundamental properties relevant to caudal propulsion, such as stiffness and movement pattern. In addition,

10 Caudal fin biorobotics Vorticit (s 1 ) Fig. 9. DPIV images, from a lateral view, of () a bluegill sunfish and () the robotic caudal fin during rolling tail motion. The fish caudal fin and the robotic caudal fin have been superimposed to show their position relative to the flow. In the flow images, ellow/red represents a positive (counterclockwise) rotation of fluid, whereas blue/purple represents a negative (clockwise) rotation of fluid. Yellow and black arrows indicate the velocit and direction of the fluid. For both images, the green regions indicate an area of zero vorticit. The flow is moving in the positive -direction. measuring forces from freel swimming fishes is challenging and subject to a number of assumptions that make interpreting these data difficult (Dabiri, 2005; Drucker and Lauder, 1999; Peng and Dabiri, 2008; Peng et al., 2007). In contrast, quantifing timedependent patterns of thrust and lift force production in a robotic device is more manageable and allows the effect of changing fin ra stiffness, frequenc and motion program on locomotor performance to be relativel easil assessed. Using an approach that investigated and modeled the caudal fin kinematics of bluegill sunfish, we developed a robotic device that was able to effectivel perform biologicall relevant motions and generate wake flows that were similar to those of a live fish (Figs 5, 9). ppropriate scaling of fin ra fleural rigidities and the careful kinematic and hdrodnamic analsis of live fish were ke to creating a robotic model that performed in a biologicall relevant manner. The robotic caudal fin ehibited a number of properties that are similar to those of the bluegill sunfish tail fin. Comparison of the fin ra curvatures and tip deflections ehibited b the robotic fin ras showed that both the curvature values and relative tip deflections were similar to those of the biological fin ras during the prescribed motions. s previousl noted, the fleural rigidities of the robotic fin ras and fish scaled equivalentl along their lengths. When the results from tracking the bending curvature and tip deflection of the biological and robotic fin ras were reviewed, it was noticed that fin ras of certain stiffness performed better than others at particular frequencies. The 500, 250 and 150 robotic fins driven at frequencies between 1.5 and 2.0 Hz resulted in tip deflection and bending curvature similar to that of the bluegill sunfish. Further eamination found that flapping the 150 at 1.5 Hz produced curvatures and tip deflections most comparable to those the sunfish. t higher stiffnesses, 1000 and 2000, the fin lacked significant curvature, due to the stiff fleural rigidities and the frequenc range tested. Furthermore, the wake flows created b the robotic caudal fin were similar to the wake patterns created b the bluegill sunfish caudal fin. oth the bluegill caudal fin and the robotic tail generated two tip vortices as the fin passed throughout the viewing area (Fig. 9). These vortices are counter-rotating and of similar vorticit. During the asmmetrical robotic motions, the vortices translated along a path on the positive -ais (dorsoventral ais) and positive -ais (rostrocaudal ais). During the cupping and W motions, the tip vortices were funneled inwards as the translated along the -ais (dorsoventral ais) (Fig. 6). This funneling effect can also be seen during in vivo bluegill locomotion (Flammang et al., 2011; Lauder, 2000; Lauder, 2006; Lauder and Madden, 2006). Components of the vorte structure created from the stead swimming motion of the bluegill sunfish also proved to be similar to those created b the robotic fin during all five motions. Effect of fin motion and shape on forces and flows ltering the motion of the fin ras, through either phase or amplitude changes, varied the forces and flows that were generated b the caudal fin robot. Using the flat motion as a control, large variations in the fin motion, such as asmmetrical motions, caused large changes in forces and flow pattern. Small variations in fin motion, such as the phase changes associated with the smmetrical motions, caused smaller changes in the force profiles. These results were consistent with those found b Zhu and Shoele (Zhu and Shoele, 2008) using CFD to predict the forces generated b fleible, trapezoidal foils. The forces and wake flows produced b the smmetrical motions were similar, but the subtle changes that were made to the fin ra motions resulted in slight, but consistent, differences. The cupping motion was able to produce significantl more thrust than the other smmetrical motions. These results are constant with previous CFD analsis that has shown that basic cupping motion might offer greater thrust production for flapping foils (Liu and ose, 1997). In addition, the cupping motion of ra-finned fishes is a common mode of movement, and has been observed in the pectoral fins of several species. In bluegill sunfish, the kinematics of pectoral fin cupping have been described (Lauder et al., 2006; Lauder et al., 2007), as have fluid flows (both computed and measured) associated with stead swimming (ozkurttas et al., 2009; Dong et al., 2010; Mittal et al., 2006). In pectoral fins, cupping motion has been demonstrated to minimize lift forces that are generated b asmmetrical fin motions, and to increase thrust on the outstroke. cupping fin motion has been observed in the caudal fin of a bluegill sunfish during stead swimming (Flammang and Lauder, 2008; Ttell, 2006), but the functional significance of the cupping motion is not full understood. The vorte structure of the cupping